We study the saddle points of the $p$-spin model—the best understood example of a “complex” (rugged) landscape—when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal{N}}$, the number of solutions averaged over randomness in the $N\to \infty$ limit. We find that it saturates the Bézout bound $ln\overline{\mathcal{N}}\sim Nln(p-1)$ The Hessian of each saddle is given by a random matrix of the form $C^{†}C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of “threshold level” well known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.

中文翻译：

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复杂的复杂景观

我们研究了 $p$自旋模型-“复杂”（崎））景观的最佳理解示例- $ñ$变量很复杂。这些是系统的解决方案$ñ$ 度数的随机方程 $p-\mathrm{1个}$。我们解决$\overline{\mathcal{ñ}}$，则解决方案的数量是在 $ñ\to \infty$限制。我们发现它饱和了Bézout边界$ln\overline{\mathcal{ñ}}〜ñln（p-\mathrm{1个}）$ 每个鞍座的Hessian由以下形式的随机矩阵给出 $C^{†}C$， 在哪里 $C$是一个复杂的对称高斯矩阵，向其对角线偏移。它的频谱在过渡过程中出现了差距，这种差距概括了在实际问题中众所周知的“阈值水平”的概念。实际问题的结果将在实际参数的限制内恢复。在这种情况下，仅解决方案总数的平方根是实数。就复杂能量而言，解决方案分为鞍形具有不同拓扑特性的部分。